Dimensions in Special Relativity Theory- a
Euclidean Interpretation*

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Dimensions in Special Relativity Theory -a Euclidean Interpretation*R.F.J. van LindenSmeetsstraat 56, 6171 VD Stein, NETHERLANDSe-mail [email protected], web http://www.euclideanrelativity.comSeptember 2005AbstractA Euclidean interpretation of special relativity is given wherein proper time τ acts as the fourthEuclidean coordinate, and time t becomes a ﬁfth Euclidean dimension. Velocity components in bothspace and time are formalized while their vector sum in four dimensions has invariant magnitude c.Classical equations are derived from this Euclidean concept. The velocity addition formula shows adeviation from the standard one; an analysis and justiﬁcation is given for that.* c Copyright Galilean Electrodynamics Vol 18 nr 1, Jan/Feb 2007. Printed with permission. PACS03.30.+p.11Introductiontime dimension. Rewriting the usual MinkowskiinvariantEuclidean relativity, both special and general, issteadily gaining attention as a viable alternativec2 = (dct/dτ )2 − (dx/dτ )2 − (dy/dτ )2 − (dz/dτ )2to the Minkowski framework, after the works of(1)a number of authors. Amongst others Montanus into Euclidean form:[1,2], Gersten [3] and Almeida [4], have paved theway. Its history goes further back, as early as 1963c2 = (cdτ /dt)2+(dx/dt)2+(dy/dt)2+(dz/dt)2 (2)when Robert d’E Atkinson [5] ﬁrst proposed Eu- one arrives at the temporal velocity componentclidean general relativity.The version in the present paper emphasizes ex-χ = cdτ /dt(3)tending the notion of velocity to the time dimen-sion. Next, the consistency of this concept in 4D This clearly deﬁnes τ as the coordinate for theEuclidean space is shown with the classical Lorentz fourth Euclidean dimension, and it says that thetransformations, after which the major inconsis- velocity components in all four dimensions involvetency with classical special relativity, the velocity derivatives with respect to t, which then can noaddition formula, is addressed.Following para- longer represent the fourth dimension. It can onlygraphs treat energy and momentum in 4D Eu- be an extra, ﬁfth dimension, x5 (provided we indexclidean space, partly using methods of relativistic the other four x1, x2, x3, and x4 respectively, withLagrangian formalism already explored by others τ = x4). This ﬁfth dimension is sometimes treatedafter which some Euclidean 4-vectors are estab- as a parameter in Euclidean approaches similar tolished.special relativity, e.g. in [1,2], but here it will beAsimpliﬁedandpopularizedversionis treated as a genuine extra Euclidean dimension. Aavailablethatwillgetyouinthe’right general expression for speed in the time dimensionmood’.It can be found on the web at (henceforth refereed to as time-speed) is now:http://www.euclideanrelativity.com.χ = cdx4/dx5(4)2The Time Dimensionwhile the scalar value of time-speed χ isχ =c2 − v2(5)Minkowski interpretations of special relativity treattime diﬀerently from spatial dimensions, showing The general expression for spatial velocity compo-from the Minkowski metric where the time compo- nents in 4D Euclidean space-time isnent is given the opposite sign. Some alternativeinterpretations (e.g. [1-4]) seek positive deﬁnitevi = dxi/dx5(6)Euclidean metrics for space-time. Also in this arti-cle, the time dimension will be treated as a regularfourth dimension in Euclidean space-time.3Using Time-Speed in SpecialIf time is considered a fourth spatial dimension,Relativitythen it must show properties similar to those foundin the other three. In there we encounter properties It will be shown that the Lorentz transformationlike length, speed, acceleration, curvature etc., ex- equations for length and time can be reproducedpressed respectively as s, ds/dt, d2s/dt2, Raetc. from the Euclidean context.bcdOf those properties, a single one can be measuredMaintaining orthogonality for all Euclidean di-relatively easily in the time dimension: the ’length’ mensions, Eqs. (2) and (5) imply that the axesor timeduration ∆t. That raises the question of for the proper time dimension and the spatial di-how a hypothetical speed in time, let us call it χ, mension in the direction of the initial motion mustshould be expressed mathematically. In [6], Greene have rotated for the moving object, as seen from thehas given a derivation of an expression that can be rest frame of the observer, in fact deﬁning Lorentzused as the velocity component in the Euclidean transformations as rotations in SO(4). See also [1],2where this is referred to as a Relative EuclideanSpace-Time. In the approach that follows now,x4x’4these axes will therefor (unlike in the Minkowski di-agram) both rotate in the same direction, clockwiseor counter clockwise, depending on the direction ofXthe motion. The diagrams in Fig. 1 and Fig. 2Cshould illustrate this.l0xVix4l’Axl4’4(i =1, 2, 3)x’Cill’xFigure 2: Object A in motion relative to observer.0AixThe dimensional axes of object A have rotated rel-i’ative to the observer.(i =1, 2, 3)• l and l4 are, respectively, the projections of thisproper length on the spatial dimensions andthe proper time dimension of the observer.Figure 1: 4D representation of an observer at OIn Fig. 2, object A moves with speed v relativeand an object A, both at rest.to the observer. This leads to a relative rotation ofdimensions x4 and xi such that V is the projectionFigure 1 depicts an object A at rest together with of the original 4D velocity C of object A on thean observer at O, also at rest. The horizontal axis xi axis of the observer at rest. The situation isshows both the spatial dimensions xexamined at the instant where xi, i = 1, 2, 3,i = xi = x4 =for the object A as well as the spatial dimensions xxi4 = 0.for the observer. The vertical axis shows both timeThe Lorentz transformation equation for x isdimensions with notation conform Eq. (2), so x4 =cτ . Due to object A being at rest, relative to thex = γ(x − vt)(7)observer, the axes overlap. The circle is just a tool whereto better show the rotation that will be depicted inγ = 1/ 1 − v2/c2(8)Fig. 2.Deﬁnitions are as follows:but this factor can also be written asγ = c/ c2 − v2 = c/χ(9)• Vector C indicates the 4D velocity, havingmagnitude c, of object A.leading tox = c(x − vt)/χ(10)• Vector V, of magnitude v, and X, of magni-tude χ, are the projections of this velocity C At t = 0, the length of object A will be contracted,on, respectively, the spatial dimensions and the as measured by the observer, according toproper time dimension of the observer.x = x χ/c(11)• l indicates the proper length of object A in the so the contraction of length l can be written asspatial direction xi in the rest frame of objectA (in this example l is also set to c).l = l χ/c(12)3which shows that l, as measured by the observerFigure 3 depicts a situation with three referenceat rest, is indeed the goniometric projection of the frames: a stationary unprimed frame x, a movingproper length l on the xi axis.primed frame x and a third, double primed frameArrow l4 is the projected ’length’ component of x of an object that moves relative to both otherthe moving object A on the proper time axis x4 frames, x and x . Each frame has dimensional axesof the observer as a result of the rotation of the rotated relative to the other frames as a result ofdimension xthe relative motion.i. This length is the manifestation ofthe diﬀerence in proper time (the non-simultaneity)between the endpoints of object A in motion ac-x4x’4cording to the Lorentz transformation equation forxtime:4’’t = γ(t − vx/c2)(13)and can be interpreted as a rotation ’out of space’of the proper length l towards the negative axis ofx0x4.At t = 0 the proper-time diﬀerence betweenV Witail and head of arrow l will beUt = −γvl/c2 = −lv/cχ(14)x’iFrom l = l χ/c and l4 = l v/c it follows thatl4 = −ct(15)which conﬁrms that lFigure 3: Relativistic addition of velocities in three4 represents the proper-timediﬀerence in object A. The factor c results from the reference frames, each with rotated dimensionalchoice of units for space and time.axes relative to each other.Summarizing, from the perspective of the ob-server, the proper length l of object A is decom-• Vector V of magnitude v is the spatial velocityposed in the components l and l4 according to:of an observer with rest frame x as measuredby an observer with rest frame x.l 2 = l2 + l24(16)• Vector W of magnitude w is the spatial veloc-and so is also the 4D speed c of the object decom-ity of a third object as measured by the ob-posed in the components χ and v:server with rest frame x.c2 = χ2 + v2.(17)• Vector U of magnitude u is the spatial velocityof that same object but now as measured byEquation (16) thus combines Eqs. (7) and (13) intothe observer with rest frame x .a single Pythagorean equation in four dimensions.When u, v, and w are parallel, the classical rela-tion between them is:4Relativistic Addition of Ve-u + vw =(18)locities1 + uv/c2If we apply the approach as used consistently untilIt appears that the Euclidean approach as used in now it yields the expression:the previous Section does not yield the same equa-1tion for relativistic addition of velocities as used inw = c cos(−α) = c sin( π + α)special relativity. Although this particular point2may be a serious obstacle to the acceptation of this= c sin(β + ϕ) = c(cos ϕ sin β + cos β sin ϕ)proposal, it obviously is necessary to point it out.= u 1 − v2/c2 + v 1 − u2/c2(19)4This expression is not nearly similar to the classicalclassical view. But if (as a matter of math-expression in Eq. (18).ematical experiment) the range of u and v isLike Eq. (18), Eq. (19) still limits the speedsextended beyond the maximum value of c thenas measured by both observers to the maximum ofthe plot looks like depicted in Fig. 5.c, which is also clear by inspection of the Figure.Some remarks will be made now on the probabilityof either of the equations to be the right one:cw1. Equation (18) is in fact based on the univer-sality of light speed and the basis for reason-ing is that an object, e.g. a photon, havingcspeed c for an observer in frame x will still havethat same speed c for an observer in frame x .u = v0This is one of Einstein’s original postulates and-calso in this Euclidean approach it will still bemaintained as a valid postulate, which essen-tially means that the photons velocity vector,as measured from the moving frame, must have-crotated along with that frame. The third ob-ject, having speed w, as measured from framex, is not a photon but a mass-carrying parti- Figure 5: Classical graph for relativistic addition ofcle for which such a rotation apparently does velocities with hypothetical (superluminal) exten-not apply. It must therefor be emphasized that sions.Eq. (19) for now may only be applied to mass-carrying particles.The part from Fig. 4 can still be recognizedbut it is clear now that this actually forms part2. Equation (18) shows a discontinuity that is un-of a continuous function that extends beyondusual in physics. In Fig. 4, Eq. (18) is plottedc. The part beyond u = v = c may not be used,for the situation where u always equals v.solely because the classical function is not de-ﬁned, nor ever shown to be valid, for such su-perluminal extensions (actually the space-likecquadrants in the classical light cone). This factstrongly suggests that the graph from Fig. 4wcis an approximation of the real function.Finally, both Eqs. (18) and (19) are plottedtogether in Fig. 6.u = v0Equation (19) is almost identical for speeds be-low about c/2 but begins to deviate at higher-cspeeds. The top of Eq. (19) corresponds to√u = v = c/ 2. From the circle diagram inFig. 3 it shows that the time-speed of the ob--cject, as measured from frame x, then becomeszero. Equation (19) further shows decreasingvalues for w in situations where the values of u√Figure 4: Graph of classical equation for relativisticand v go beyond c/ 2 (the frame of the mov-addition of velocities.ing object then rotates beyond π/2 relative toframe x). It turns out that in that case the cor-With u and v nearing c, the resulting w willresponding time-speed for the object becomesalso near c, which is in accordance with thenegative. (This situation might be related to5ccNeww-wcClassic-cClassicNew-c/ 2u = v00u = v/3-0.949ccc-c-cFigure 6: Classical [Eq. (18)] and newly derived Figure 7: Classical [Eq. (18)] and newly derivedgraph [Eq. (19)] for relativistic addition of veloci- graph [Eq. (19)] for relativistic addition of veloci-ties plotted together for u = v.ties plotted together for u = v/3.anti-particles, running ’backwards in time’.).on the order of 10−5 m/s, which might be no-The situation where u equals v gives the max-ticeable using adequately accurate measuringimum possible deviation relative to the clas-devices.sical graph. Other ratios between u and vgive (much) smaller deviations and the topsA hypothetical case will now be used to showof Eq. (19) will shift outwards towards c as that Eq. (19) does not necessarily lead to causalitycan be seen in Fig. 7 where the ratio between conﬂicts as a result of the negative time-speeds thatu and v equals 3:1. At a ratio 10:1 both plots can occur.are practically identical. Virtually all practi-A spaceship travels relative to Earth at speedcal situations that require the velocity addition vs = 0.9c and heads toward an asteroid that is atformula to be used exist under such circum- rest relative to Earth. The ship launches a missilestances, which indicates that a deviation from at the asteroid at vm = 0.9c relative to the ship.the classical graph is likely to remain unno- An observer on the ship watches the missile destroyticed.the asteroid. According to Eq. (19), an observeron Earth would see the missile traveling at only3. Some interpretations of Fizeau’s experiment 0.7846c so the missile’s spatial speed is lower thangive rise to doubt concerning the correctness that of the spaceship. It seems therefor that thisof Eq. (18). If Eq. (19) is used in the anal- observer would see the ship hit the asteroid beforeysis of Fizeau’s experiment done by Renshaw the missile.[7], it yields better results than Eq. (18), al-The explanation of this paradox can be found inthough still not within the margins as claimed the comparison of the proper times of all objectsby Michelson.involved. We call the proper time for the spaceshipThe vast majority of experimental set-ups that τs and for the missile τm. For simplicity we setare aimed at veriﬁcation of relativity theory the space-time event of the launch at t = τm =are using two reference frames. These exper- τs = 0 and the distance between the spaceship andiments are not suitable for the veriﬁcation of the asteroid at that moment at 0.9 light second (asthe velocity addition formula. One would have measured by the observer on Earth).to use a set-up with three reference frames. AtTheobserveronEarthcalculatestime-speeds on the order of 104 m/s the diﬀerence in coordinates of the impact (against the asteroid)resulting values between Eqs. (18) and (19) is using his own time t for the spaceship: ts = 1s;6and for the missile: tm = 0.9/0.7846 = 1.147s, so itThe relativistic Doppler eﬀect can thus be inter-seems as if the spaceship reaches the asteroid ﬁrst. preted as a combination of the normal ’acoustic’In 4D Euclidean space-time however the observer Doppler eﬀect in space and a frequency shift thatmeasures the time-speed χs of the spaceship as: results from the lower time-speed.χs =c2 − v2s =c2 − (0.9c)2 = 0.4359c.According to this observer the absolute valueof the timespeed χ6Mass, Energy and Momen-m of the missile is χm=c2 − (0.7846c)2 = 0.62c, but from the circle di-tumagram (Fig. 3) it shows that we must now takethe negative root so its value is χm = −0.62c. Figure 8 depicts a moving object with spatial veloc-Note that the cyclic nature of γ now also implies ity V of magnitude v, as measured by an observerthat in this situation γ has a negative value in at point L, at rest.τm = tm/γ = tmχm/c for the missile.We calculate the proper times at the momentxof impact according to the observer on Earth for4x’4the spaceship: τMs = tsχs/c = 0.4359s; and for the’missile: τKm = 1.147(−0.62) = −0.7111s.KIn proper time the missile hits the asteroid beforeXCthe spaceship does despite its lower spatial speed.Causality is therefor not violated. The missile runs0backwards in proper time.xLiV5Relativistic Doppler Eﬀect√Using the identity χ =c2 − v2 for the time-speedvariable in the wavelength equation for the rela-tivistic Doppler eﬀect1 + v/cFigure 8: 4D velocity of magnitude c in x4 of anλ = λ0(20)1 − v/cobject at L. An observer at rest at L has velocityof magnitude c in x4.simpliﬁes this expression toThe vector sum of spatial and time-velocities re-λ = λ0(c + v)/χ(21) ﬂects the four-velocities of the observer (along x4)It is possible to identify the individual contribu- and the moving object (along x4). It follows natu-tions of the factors v and χ to the total Doppler rally that the Lorentz invariant m0c (m0 is the resteﬀect by considering χ = c (which isolates the ef- mass) in the moving object A can be decomposedfect of the spatial speed) and v = 0 (which isolates inthe eﬀect of the time-speed).m20c2 = m20χ2 + m20v2(24)Setting χ = c results in:which, using the identities E = γm0c2 and p =γmλ0v, is equivalent to the classical equationv = λ0(1 + v/c)(22)E2/c2 = m2which is the regular equation for the acoustic0c2 + p2(25)Doppler eﬀect with moving source and stationary E being the total energy and p being the spatialreceiver. Setting v = 0 results in:momentum.λThe components in the right part of Eq. (24)χ = λ0c/χ(23) cannot simply be interpreted as, respectively, thewhich simply makes the photon’s frequency propor- object’s momenta in the time dimension and thetional to the time-speed of the emitting particle.spatial dimension of the rest frame of the observer.7There is an obvious problem in the fact that the which equals, as a result of the universal velocityfactor γ is involved in the expressions for E and p. magnitude c for the free particle in 4D space-time:If we multiply the factor γ2 into all three elementsof Eq. (24) we get:Λ = m0c2(30)γ2m2The latter is to be interpreted as the ’kinetic en-0c2 = γ2m20χ2 + γ2m20v2(26) ergy’ of the particle in four dimensions, which iswhich describes triangle LK’M (if ma fundamentally diﬀerent concept than kinetic en-0 is set to 1).This alternative form for Eq. (24) immediately ergy in three dimensions. It corresponds to theshows the meaning of its components. They now total energy of a particle at rest. Other solutionscorrespond one to one with the components in Eq. for Λ are possible but the essential element is that(25): γmany solution is a constant in 4D space-time.0c = E/c, γm0χ = m0c, γm0v = p. Thefactor γmThe relativistic Lagrangian Λ shows that the fac-0c is however not invariant under rota-tions in SO(4), while mtor γ in Eq. (26) must be a result of our conﬁne-0c is. [Note that althoughmment to a 3D subspace of 4D space-time. In order0c is indeed Lorentz invariant from the perspec-tive of the observer, its physical meaning in its own to maintain conservation laws for energy and mo-rest frame is the moving object’s time-momentum. mentum, while only being able to measure theirThe same invariant value can be found in the rest ’projections’ to our 3D space, the factor γ is anframe of the observer (see also Fig. 9) but should artiﬁcial necessity. It vanishes for a hypotheticalthen be read as γmobserver with full 4D observational skills, who mea-0χ.]The Lagrangian formal-ism for this situation has been worked out by Mon- sures the object’s speed and energy as constants.tanus in [2]. The reader is therefore referred tothis source for the detailed derivation. The generic 7 Transformation of Energyprinciples used for such 5D situations (or more gen-erally 4D with the addition of an extra parameterand Momentumto keep track of the progress of the object along itsworld-line) appear in Goldstein [8]. The latter how- The generic transformation equations for energyever uses the classical indeﬁnite Minkowski metric and momentum depend indirectly on the equationas a basis for the development of the relativistic La- for relativistic addition of velocities. Because a newgrangian Λ where Montanus uses a positive deﬁnite one replaces this equation, it is necessary to reworkmetric like in this article. A short overview of the the transformation equations for energy and mo-main equations is given here.mentum as well.In agreement with classical mechanics it is as-Figure 9 depicts an object moving with velocitysumed that the variation according to Hamilton’s W of magnitude w relative to frame x and velocityprinciple:U of magnitude u relative to frame x .(please refer also to Fig. 3 and the deﬁnitionsx5(2)given there)δI =Λ(xµ, uµ)dx5(27)x5(1)• E = γ(w)m0c2 is the energy of an object thatmoves with velocity W of magnitude w relativeis an extremum, where uµ = dxµ/dx5. The corre-to frame x and measured in frame x.sponding Euler-Lagrange equations of motion are:• E = γ(u)m∂Λd0c2 is the energy of that same ob-−(∂Λ/∂uject moving with velocity U of magnitude u∂xµ) = 0(28)µdx5relative to frame x and measured from framex .leading to a possible relativistic Lagrangian for afree object in the absence of a forceﬁeld (so the• Frame x moves with velocity V of magnitudepotential energy equals zero):v relative to frame x.Λ = m• γ(u) = 11 − u2/c20uµuµ(29)8xwhere ds = cdτ . Four-vectors with the Euclidean4metric (+1, +1, +1, +1) as used in the previous Sec-x’4Ec/c = (w)m c(w)m= m000tions use the 4D velocity of the moving object and4D Euclidean distances as invariants, which is inx4’’fact the essence of Eq. (2):XE /c’ = (u)m c0c2 = v21 + v22 + v23 + χ2(35)0xV WiUMultiplication with dt2 = dx2Note:5 yields (recall thatm =1χ = cdτ /dt):0x’ic2dt2 = dx21 + dx22 + dx23 + c2dτ2(36)where the factors c2dτ 2 and c2dt2 from Eq. (34)have switched roles.Figure 9: Generic transformation of energy and mo-The Euclidean metric thus gives rise tomentum in three reference frames with rotated di- four-vectorsforposition,velocityanden-mensional axes.ergy/momentum:EuclideanMinkowskian• γ(v) = 11 − v2/c2(x1, x2, x3, cτ)(x1, x2, x3, ct)(v1, v2, v3, χ)γ(v1, v2, v3, c)• γ(w) = 11 − w2/c2(m0v1, m0v2, m0v3, m0χ)(p1, p2, p3, E/c)For energy this leads to a generic transformationEquation (36) is not really new. It is merely Eq.equation(34) written in a diﬀerent form, with as a mainE/E = γ(w)/γ(u)(31) input the deﬁnition of χ, being the time-speed ofwhich can be written in diﬀerent forms using Eq. an object as measured by an observer at rest, which(19). With u = 0 this reduces to the classical form: has three eﬀects:E/E = γ(v)(32)• It creates a new invariant c, being the universalmagnitude of the 4D velocity of an object.For momentum a generic transformation equationis• It provides a Euclidean basis for the deﬁnitionp/p = wE/uE(33)of vectors in the direction of the time dimen-sion.where:• It enables these new vectors to be summed• p = γ(u)m0u is the momentum of the objectwith existing vectors in the spatial dimensions.as measured from frame x .• p = γ(w)mIn general, the new Euclidean four-vectors can be0w is the momentum of the objectas measured from frame x.derived from the Minkowski four-vectors by usingthe time component in the Minkowski four-vectoras the invariant (the vector sum) for the new four-8Euclidean Four-Vectorsvector. It is essentially doing Pythagoras “the otherway around”, i.e., calculating the hypotenuse fromThe traditional Minkowski line element with metric the rectangular sides, instead of calculating a rect-(+1, −1, −1, −1) is:angular side from the hypotenuse and the otherrectangular side (refer to [9] for a detailed treat-ds2 = c2dt2 − dx2 − dy2 − dz2(34) ment of Minkowski and Euclidean four-vectors).9References[1] H. Montanus, ”Proper-Time Formulation ofRelativistic Dynamics”, Foundations of Physics31 (9) 1357-1400 (2001).[2] H. Montanus, ”Special Relativity in an Abso-lute Euclidean Space-Time”, Physics Essays 4(3) 350-356 (1991).[3] A. Gersten, ”Euclidean special relativity”, Foun-dations of Physics 33 (8) 1237-1251 (2003).[4] Jose B. Almeida, ”An Alternative to MinkowskiSpace-Time” (arXiv:gr-qc/0104029 v2, 10 Jun2001).[5] Robert d’E Atkinson, ”General Relativity inEuclidean Terms”, Royal Society of LondonProceedings Series A 272 (1348) 60-78 (1963).[6] Brian R. Greene, The elegant universe, page391 note 5 (W.W. Norton & Company NewYork-London, 1999).[7] C. Renshaw, ”The Experiment of Fizeau as aTest of Relativistic Simultaneity”, available at:renshaw.teleinc.com.[8] H. Goldstein, Classical Mechanics, Secondedition, Chapter 7-9 (Addison Wesley, 1980).[9] R.F.J.vanLinden”Minkowskiver-susEuclidean4-vectors”,availableat:www.euclideanrelativity.com.10

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Dimensions in Special Relativity Theory- a Euclidean Interpretation*

An Euclidean interpretation of special relativity is given wherein proper time acts as the fourth Euclidean coordinate, and time t becomes a fifth Euclidean dimension. Velocity components in both space and time are formalized while their vector sum in four dimensions has invariant magnitude c . Classical equations are derived from this Euclidean concept. The velocity addition formula shows a deviation from the standard one; an analysis and justification is given for that.

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